Mathematical Basics. Chapter Introduction and Definitions

Transcription

1 Chapter 2 Mathematical Basics 2.1 Introuction an Definitions Flui mechanics eals with transport processes, especially with the flow- an molecule-epenent momentum transports in fluis. Their thermoynamic properties of state such as pressure, ensity, temperature an internal energy enter into flui mechanics consierations. The thermoynamic properties of state of a flui are scalars an as such can be introuce into the equations for the mathematical escription of flui flows. However, in aition to scalars, other kins of quantities are also require for the escription of flui flows. In the following sections it will be shown that flui mechanics consierations result in conservation equations for mass, momentum, energy an chemical species which comprise scalar, vector an other tensor quantities. Often funamental ifferentiations are mae between such quantities, without consiering that the quantities can all be escribe as tensors of ifferent orers. Hence one can write: Scalar quantities = tensors of zero orer {a} a Vectorial quantities = tensors of first orer {a i } a i Tensorial quantities = tensors of secon orer {a ij } a ij where the number of the chosen inices i, j, k, l, m, n of the tensor presentation esignatesthe oreran a can be any quantity uner consieration. The introuction of tensorial quantities, as inicate above, permits extensions of the escription of flui flows by means of still more complex quantities, such as tensors of thir or even higher orer, if this becomes necessary for the escription of flui mechanics phenomena. This possibility of extension an the above-mentione stanar escriptions le us to choose the inicate tensor notation of physical quantities in this book, the number of the inices i, j, k, l, m, n eciing the orer of a consiere tensor. Tensors of arbitrary orer are mathematical quantities, escribing physical properties of fluis, with which mathematical operations such as aition, 15

2 16 2 Mathematical Basics subtraction, multiplication an ivision can be carrie out. These may be well known to many reaers of this book, but are presente again below as a summary. Where the brevity of the escription oes not make it possible for reaers, not accustome to tensor escriptions, to familiarize themselves with the matter, reference is mae to the corresponing mathematical literature; see Sect Many of the following euctions an escriptions can, however, be consiere as simple an basic knowlege of mathematics an it is not necessary that the etails of the complete tensor calculus are known. In the present book, only the tensor notation is use, along with simple parts of the tensor calculus. This will become clear from the following explanations. There are a number of books available that eal with the matter in the sections to come in a mathematical way, e.g. see refs. [2.1] to [2.7]. 2.2 Tensors of Zero Orer (Scalars) Scalars are employe for the escription of the thermoynamic state variables of fluis such as pressure, ensity, temperature an internal energy, or they escribe other physical properties that can be given clearly by stating an amount of the quantity an a imensional unit. The following examples explain this: [ ] [ ] [ ] N kg P =7.53 }{{ 10 } 6 Amount m 2 }{{} Unit, T = } {{} Amount K }{{} Unit, ρ =1.5 } {{ 10 } 3 Amount m 3 }{{} Unit (2.1) Physical quantities that have the same imension can be ae an subtracte, the amounts being inclue in the aing an subtracting operations, with the common imension being maintaine: N a α = α=1 N a α }{{} α=1 Amount [ ] a }{{} Unit [ ] [ ] [ ] a±b =( a ± b ) a or b, with a = b }{{}}{{} Amount Unit (2.2) Quantities with iffering imensions cannot be ae or subtracte. The mathematical laws below can be applie to the permitte aitions an subtractions of scalars, see for etails [2.5] an [2.6]. The amount of a isarealnumber,i.e. a is a real number if a R. Itis efine by a := +a, if a 0an a := a, if a<0. The following mathematical rules can be eucte irectly from this efinition: a a a, a = a, ab = a b, a = a (if b 0) b b a b b a b

3 2.3 Tensors of First Orer (Vectors) 17 From a a a an b b b, it follows that ( a + b ) a + b ( a + b ). Thus for all a, b R: a + b a + b (triangular inequality) The commutative an associative laws of aition an multiplication of scalar quantities are generally known an nee not be ealt with here any further. If one carries out multiplications or ivisions with scalar physical quantities, new physical quantities are create. These are again scalars, with amounts that result from the multiplication or ivision of the corresponing amounts of the initial quantities. The imension of the new scalar physical quantities results from the multiplication or ivision of the basic units of the scalar quantities: [ ] a b =( a b ) [a] [b] }{{}}{{} Amount Unit an a b = a [ ] [a] b [b] }{{}}{{} Amount Unit (2.3) It can be seen from the example of the prouct of the pressure P an the volume V how a new physical quantity results: [ ] [ ] N P V = P V m 2 m3 = P V Nm (2.4) }{{} [J=N m] The new physical quantity has the unit J = joule, i.e. the unit of energy. When a pressure loss P is multiplie with the volumetric flow rate, a power loss results: [ ] [ ] P V = P V N m 2 m3 = P s V Nm (2.5) s }{{} [W= Nm s ] The power loss has the unit W = watt = joule/s. 2.3 Tensors of First Orer (Vectors) The complete presentation of a vectorial quantity requires the amount of the quantity to be given, in aition to its irection an its unit. Force, velocity, momentum, angular momentum, etc., are examples for vectorial quantities. Graphically, vectors are represente by arrows, whose length inicates the amount an the position of the arrow origin an the arrowhea inicates the irection. The erivable analytical escription of vectorial quantities makes use of the inication of a vector component projecte on to the axis of a coorinate system, an the inication of the irection is shown by the signs of the resulting vector components.

4 18 2 Mathematical Basics Fig. 2.1 Representation of velocity vector U i in a Cartesian coorinate system To represent the velocity vector {U i }, for example, in a Cartesian coorinate system, the components U i (i =1, 2, 3) can be expresse as follows: U 1 cos α 1 [ m ] [ m ] U = {U i } = U 2 = U cos α 2 U i = ± U 3 cos α s }{{} U cos α i }{{} s 3 Direction }{{} Amount Unit (2.6) Looking at Fig. 2.1, one can see that the following hols: U 1 = U 1 e 1, U 2 = U 2 e 2, U 3 = U 3 e 3 (2.7) where the unit vectors e 1, e 2, e 3 in the coorinate irections x 1,x 2 an x 3 are employe. This is shown in Fig α i esignates the angle between U an the unit vector e i. Vectors can also be represente in other coorinate systems; through this, the vector oes not change in itself but its mathematical representation changes. In this book, Cartesian coorinates are preferre for presenting vector quantities. Vector quantities which have the same unit can be ae or subtracte vectorially. Laws are applie here that result in aition or subtraction of the components on the axes of a Cartesian coorinate system: a ± b = {a i }±{b i } = {(a i ± b i )} = {(a 1 ± b 1 ), (a 2 ± b 2 ), (a 3 ± b 3 )} T Vectorial quantities with ifferent units cannot be ae or subtracte vectorially. For the aition an subtraction of vectorial constants (having the same units), the following rules of aition hol: a + 0 = {a i } + {0} = a (zero vector or neutral element 0) a +( a) ={a i } + { a i } = 0 (a element inverse to a) a + b = b + a,.h. {a i } + {b i } = {b i } + {a i } = {(a i + b i )} (commutative law) a +(b + c) =(a + b)+c,.h. {a i } + {(b i + c i )} = {(a i + b i )} + {c i } (associative law)

5 2.3 Tensors of First Orer (Vectors) 19 With (α a) a multiple of a results, if α>0. α has no unit of its own, i.e. (α a) esignates the vector that has the same irection as a but has α times the amount. In the case α<0, one puts (α a) := ( α a). For α =0the zero vector results: 0 a = 0. When multiplying two vectors two possibilities shoul be istinguishe yieling ifferent results. The scalar prouct a b of the vectors a an b is efine as { a b cos(a, b), if a 0 an b 0 a b := (2.8) 0, if a = 0 or b = 0 where the following mathematical rules hol: a b = b a a b = 0 if a orthogonal to b (αa) b = a (αb) =α(a b) a ef = a a (a + b) c = a c + b c (2.9) When the vectors a an b are represente in a Cartesian coorinate system, the following simple rules arise for the scalar prouct (a b) anforcos(a, b): a b = a 1 b 1 + a 2 b 2 + a 3 b 3, a = a a2 2 + a2 3 (2.10) cos(a, b) = a b a b = a 1 b 1 + a 2 b 2 + a 3 b 3 a a a2 3 b b (2.11) b2 3 The above equations hol for a, b 0. Especially the irectional cosines in a Cartesian coorinate system are calculate as a i cos(a, e i )= a a i =1, 2, 3 (2.12) a2 3 i.e. a i represents the angles between the vector a an the base vectors e 1 = 0, e 2 = 1, e 3 = 0 (2.13) The vector prouct a b of the vectors a an b has the following properties: a b is a vector 0, ifa 0 an b 0 an a is not parallel to b; a b = a b sin(a, b) (area of the parallelogram set up by a an b); a b is a vector staning perpenicular to a an b an can be represente with (a, b, a b), a right-hane system. It can easily be seen that a b = 0, ifa = 0 or b = 0 or a is parallel to b. One shoul take into consieration that for the vector prouct the associative law oes not hol in general: a (b c) (a b) c

6 20 2 Mathematical Basics Fig. 2.2 Graphical representation of a vector prouct a b The following computation rules can be state (Fig. 2.2): a a =0, a b = (b a), α(a b) =(αa) b = a (αb) (for α R) a (b + c) =a b + a c (a + b) c = a c + b c (istributive laws) a b =0 a =0orb =0ora, b parallel (parallelism test) a b 2 = a 2 b 2 (a b) 2 If one represents the vectors a an b in a Cartesian coorinate system with e i, the following computation rule results: a 1 b 1 a 2 b 2 = e 1 a 1 b 1 a 2 b 3 a 3 b 2 e 2 a 2 b 2 a 3 b 3 e 3 a 3 b 3 = a 3 b 1 a 1 b 3 (2.14) a 1 b 2 a 2 b 1 The tensor of thir orer ɛ ijk := e i (e j e k ) that will be introuce in Sect. 2.5 permits, moreover, a computation of a vector prouct accoring to {a i } {b j } := ɛ ijk a i b j (2.15) A combination of the scalar prouct an the vector prouct leas to the scalar triple prouct (STP) forme of three vectors: [ ] a, b, c = a (b c) (2.16) The properties of this prouct from three vectors can be seen from Fig The STP of the vectors a, b, c leas to six times the volume of the parallelopipe (pp), V pp, efine by the vectors: a, b an c. The parallelopipe prouct of the three vectors a, b, c is calculate from the value of a triple-row eterminant: a 1 b 1 c 1 [a, b, c] = a 2 b 2 c 2 (2.17) a 3 b 3 c 3

7 2.4 Tensors of Secon Orer 21 Fig. 2.3 Graphical representation of scalar triple prouct by three vectors V pp = 1 6 V STP = 1 [a, b, c] (2.18) 6 It is easy to show that for the STP a (b c) =b (a c) =c (a b) b (a c) (2.19) For the vector triple prouct a b c, the following relation hols: a (b c) =(a c)b (a b)c (2.20) Further important references are given in books on vector analysis; see also Sect Tensors of Secon Orer In the preceing two sections, tensors of zero orer (scalar quantities) an tensors of first orer (vectorial quantities) were introuce. In this section, a summary concerning tensors of secon orer is given, which can be formulate as matrices with nine elements: a 11 a 12 a 13 {a ij } = a 21 a 22 a 23 = a ij (2.21) a 31 a 32 a 33 In the matrix element a ij, the inex i represents the number of the row an j represents the number of the column, an the elements esignate with i = j are referre to as the iagonal elements of the matrix. A tensor of secon orer is calle symmetrical when a ij = a ji hols. The unit secon-orer tensor is expresse by the Kronecker elta: 100 δ ij = 010, i.e. δ ij = 001 { +1 if i = j 0 if i j (2.22)

8 22 2 Mathematical Basics The transpose tensor of {a ij } is forme by exchanging the rows an columns of the tensor: {a ij } T = {a ji }. When oing so, it is apparent that the transpose unit tensor of secon orer is again the unit tensor, i.e. δij T = δ ij. The sum or ifference of two tensors of secon orer is efine as a tensor of secon orer whose elements are forme from the sum or ifference of the corresponing ij elements of the initial tensors: a 11 ± b 11 a 12 ± b 12 a 13 ± b 13 {a ij ± b ij } = a 21 ± b 21 a 22 ± b 22 a 23 ± b 23 (2.23) a 31 ± b 31 a 32 ± b 32 a 33 ± b 33 In the case of the following presentation of tensor proucts, often the socalle Einstein s summation convention is applie. By this one unerstans the summation over the same inices in a prouct. When forming a prouct from tensors, one istinguishes the outer prouct an the inner prouct. The outer prouct is again a tensor, where each element of the first tensor multiplie with each element of the secon tensor results in an element of the new tensor. Thus the prouct of a scalar an a tensor of secon orer forms a tensor of secon orer, where each element results from the initial tensor of secon orer by scalar multiplication: α a 11 α a 12 α a 13 α {a ij } = {α a ij } = α a 21 α a 22 α a 23 (2.24) α a 31 α a 32 α a 33 The outer prouct of a vector (tensor of first orer) an a tensor of secon orer results in a tensor of thir orer with altogether 27 elements. The inner prouct of tensors, however, can result in a contraction of the orer. As examples are cite the proucts a ij b j : a 11 a 12 a 13 b 1 a 11 b 1 + a 12 b 2 + a 13 b 3 {a ij } {b j } = a 21 a 22 a 23 b 2 = a 21 b 1 + a 22 b 2 + a 23 b 3 (2.25) a 31 a 32 a 33 b 3 a 31 b 1 + a 32 b 2 + a 33 b 3 an T a 11 a 12 a 13 b 1 a 11 + b 2 a 21 + b 3 a 31 {b i } T {a ij } = {b 1,b 2,b 3 } a 21 a 22 a 23 = b 1 a 12 + b 2 a 22 + b 3 a 32 a 31 a 32 a 33 b 1 a 13 + b 2 a 23 + b 3 a 33 (2.26) In summary, this can be written as {a ij } {b j } = {(a ij b j )} = {(ab) i } (2.27) an {b i } {a ij } = {(b i a ij )} = {(ab) j } (2.28)

9 2.5 Fiel Variables an Mathematical Operations 23 If one takes into account the above prouct laws: {δ ij } {b j } = {b i } an {b i } T {δ ij } = {b j } T (2.29) The multiplication of a tensor of secon orer by the unit tensor of secon orer, i.e. the Kronecker elta, yiels the initial tensor of secon orer: 100 a 11 a 12 a 13 {δ ij } {a ij } = 010 a 21 a 22 a 23 = {a ij} (2.30) 001 a 31 a 32 a 33 Further proucts can be formulate, as for example cross proucts between vectors an tensors of secon orer: {a i } {b jk } = ɛ ikj a i b jk (2.31) but these are not of special importance for the erivations of the basic laws in flui mechanics. 2.5 Fiel Variables an Mathematical Operations In flui mechanics, it is usual to present thermoynamic state quantities of fluis, such as ensity ρ, pressure P, temperature T an internal energy e, as a function of space an time, a Cartesian coorinate system being applie here generally. To each point P(x 1,x 2,x 3 )=P(x i )avalue ρ(x i,t),p(x i,t),t(x i,t), e(x i,t), etc., is assigne, i.e. the entire flui properties are presente as fiel variables an are thus functions of space an time Fig It is assume that in each point in space the thermoynamic connections between the state quantities hol, as for example the state equations that can be formulate for thermoynamically ieal fluis as follows: ρ = constant P/ρ = RT (state equation of the thermoynamically ieal liquis) (state equation of the thermoynamically ieal gases) Entirely analogous to this, the properties of the flows can be escribe by introucing the velocity vector, i.e. its components, as functions of space an time, i.e. as vector fiel Fig Furthermore, the local rotation of the flow fiel can be inclue as a fiel quantity, as well as the mass forces an mass acceleration acting locally on the flui. Thus the velocity U j = U j (x i,t), the rotation ω j = ω j (x i,t), the force K j = K j (x i,t) an the acceleration g j (x i,t) canbestateasfielquantitiesancanbeemployeassuchquantitiesin the following consierations. In an analogous manner, tensors of secon an higher orer can also be introuce as fiel variables, for example, τ ij (x i,t), which is the moleculeepenent momentum transport existing at a point in space, i.e. at the point

10 24 2 Mathematical Basics r Fig. 2.4 Scalar fiels assign a scalar to each point in the space an as a function of time Fig. 2.5 Vector fiels assign vectors to each point in the space as functions of time P(x i )attimet. Itrepresentsthej-momentum transport acting in the x i irection. Further, ɛ ij (x i,t) represents the flui element eformation epening on the graients of the velocity fiel at the location P(x i )attimet. The properties introuce as fiel variables into the above consierations represente tensors of zero orer (scalars), tensors of first orer (vectors) an tensors of secon orer. They are employe in flui mechanics to escribe flui flows an the corresponing flui escription is usually attribute to Euler ( ). In this escription, all quantities consiere in the representations of flui mechanics are ealt with as functions of space an time. Mathematical operations such as aition, subtraction, ivision, multiplication, ifferentiation an integration, that are applie to these quantities, are subject to the known laws of mathematics. The ifferentiation of a scalar fiel, for example the ensity ρ(x i,t), gives ρ t = ρ t + ρ x 1 ( x1 t = ρ 3 ( ρ t + i=1 x i ) + ρ x 2 ( x2 t ) + ρ x 3 )( ) xi = ρ ( ρ t t + x i ( x3 t ) )( ) xi t (2.32) In the last term, the summation symbol 3 i=1 was omitte an the Einstein s summation convention was employe, accoring to which the ouble inex

11 2.5 Fiel Variables an Mathematical Operations 25 ( )( ) i in ρ xi x i t prescribes a summation over three terms i =1, 2, 3, i.e.: 3 ( ρ i=1 x i )( xi t ) ( ρ = x i )( xi t ) (2.33) The ifferentiation of vectors is given by the following expressions: { } T U t = U1 t, U 2 t, U 3 U i, i =1, 2, 3 (2.34) t t i.e. each component of the vector is inclue in the ifferentiation. As the consiere velocity vector epens on the space location x i an the time t, the following ifferentiation law hols: U j = U j t t + U ( ) j xi (2.35) x i t When one applies the Nabla or Del operator: { } T { } =,, =, x 1 x 2 x 3 x i i =1, 2, 3 (2.36) on a scalar fiel quantity, a vector results: a = { a x 1, a x 2, } T { } a a = gra a =, i =1, 2, 3 (2.37) x 3 x i This shows that the Nabla or Del operator results in a vector fiel euce from the graient fiel. The ifferent components of the resulting vector are forme from the prevailing partial ifferentiations of the scalar fiel in the irections x i. The scalar prouct of the operator with a vector yiels a scalar quantity, i.e. when ( ) applie to a vector quantity results in: a = a 1 + a 2 + a 3 = iv a = a i (2.38) x 1 x 2 x 3 x i Here, in a i / x i the subscript i again inicates summation over all three terms, i.e. 3 i=1 a i x i = a i x i (Einstein s summation convention) (2.39) The vector prouct of the operator with the vector a yiels corresponingly e 1 / x 1 a 1 a 3 / x 2 a 2 / x 3 a = e 2 / x 2 a 2 e 3 / x 3 a 3 = a 1 / x 3 a 3 / x 1 = rot a (2.40) a 2 / x 1 a 1 / x 2

12 26 2 Mathematical Basics or a = rot a = ɛ ijk a i x j = ɛ ijk a j x i (2.41) The Levi Civita symbol ɛ ijk is also calle the alternating unit tensor an is efine as follows: ɛ ijk = { 0 : if two of the three inices are equal +1 : if ijk = 123, 231 or :if ijk = 132, 213 or 321 (2.42) Concerning the above-mentione proucts of the operator, the istributive law hols, but not the commutative an associative laws. If one applies the operator to the graient fiel of a scalar function, the Laplace operator 2 (alternative notation ) results. When applie to a, the result can be written as follows: 2 a =( )a = 2 a x a 1 x a 2 x 2 = 2 a (2.43) 3 x i x i The Laplace operator can also be applie to vector fiels, i.e. to the components of the vector: ( ) ( ) ( 2 U 1 2 U 1 / x U 1 / x U 1 / x 2 2 U = 2 ( ) ( ) ( 3) U 2 = 2 U 2 / x U 2 / x U 2 2 3) (2.44) ( ) ( ) ( 2 U 3 2 U 3 / x U 3 / x U 3 / x3) Substantial Quantities an Substantial Derivative A further approach to escribing flui mechanics processes is to erive the basic equations in terms of substantial quantities. This approach is generally name after Lagrange ( ) an is base on consierations of properties of flui elements. The state quantities of a flui element R such as the ensity ρ R, the pressure P R, the temperature T R an the energy e R are employe for the erivation of the laws of flui motion. If one wants to measure or escribe these properties of a flui element in a fiel, one has to move with the element, i.e. one has to follow the path of the element: (x i ) R,T =(x i ) R,0 + T 0 (U i ) R t (2.45) As the path of a flui element is only a function of time t an an initial space coorinate (x i ) R,0, the substantial quantities, i.e. the thermoynamic state quantities of a flui element can also only be functions of time.

13 2.7 Graient, Divergence, Rotation an Laplace Operators 27 Thus the total ifferentials of all substantial quantities can be formulate as follows, with reference to (2.35): a R = a t t + a ( ) ( ) xi xi, where =(U i ) R (2.46) x i t t R R The quantity specifie with (x i /t) R inicates the change of position of a flui element R with time, i.e. the substantial velocity of a flui element. If a flui element is positione at time t at location x i,then(u i ) R = U i results an from this arises the final equation of the substantial erivative of afielvariableda/dt: a R t = Da Dt = a t + U a i (2.47) x i This equation results also from the ientity relationship. This states, for a representation of a flui flow in two ifferent ways, without contraiction, that the following equality of Euler an Lagrange variables hols: a R (t) =a(x i,t)if(x i ) R = x i at time t From this one can erive a R t = a t +(U a i) R = a x i t + U i a = Da x i Dt (2.48) From (2.47), it can be seen that the operator D Dt (= substantial erivative) can be written as follows: D Dt = t + U i = +(U ) (2.49) x i t This operator can be applie to fiel variables an is very important for the subsequent erivations of the basic equations of flui mechanics, as it permits the formulation of the basic equations in Lagrange variables an in a secon step the subsequent transformation of all terms in this equation into Euler variables. In this final form, i.e. expresse in Euler variables, the equations are suite for the solution of practical flow problems. 2.7 Graient, Divergence, Rotation an Laplace Operators a(x i,t) represents a scalar fiel, i.e. it is efine or given as a function of space an time. The graient fiel of the scalar a can be assigne the following components at each point in a space:

14 28 2 Mathematical Basics { } a a/ x 1 a(x i,t)= =gra(a) = a/ x 2 x i ; gra(a) =f(x i,t) (2.50) a/ x 3 Thus the operator gra( ) is efine as follows: { } { () () gra() = = x i x 1 () x 2 } () x 3 (2.51) i.e. gra(a) is a vector fiel, whose components are marke by the inex i. The gra(a) vectors exhibit irections which are perpenicular to the lines of a = constant of the consiere scalar fiel, i.e. perpenicular to a(x i,t)= constant. Furthermore, the Laplace operator can be assigne to each scalar fiel a(x i,t) a(x i,t) (J.S. Laplace ( )). Here, a(x i,t) is a scalar fiel, e.g. to each space point the quantity (a) is assigne a(x i,t)= 2 a = 2 a x i x i x a 1 x a 2 x 2 (2.52) 3 Employing the previously efine ivergence operator, the following equation results: ( ) a a(x i,t) = iv(gra a) =iv = 2 a = 2 a x i x i x i x 2 (2.53) i For the mathematical treatment of flow problems, there are other mathematical operators of importance, in aition to the operators iv( ) an rot( ) = curl( ) that are applicable to vector fiels such as U(x i,t) an are efine as follows: an iv (U(x j,t)) = U i x i = U 1 x 1 + U 2 x 2 + U 3 x 3 (2.54) rot U(x j,t)=ɛ ijk U U 3 / x 2 U 2 / x 3 j = U 1 / x 3 U 3 / x 1 x i U 2 / x 1 U 1 / x 2 (2.55) Here, iv U is a scalar fiel an rot or curl U are vector fiels. When U is a velocity fiel, the value of iv U escribes the temporal change of the volume δv R of a flui element with constant mass δm R, i.e. iv (U) = U i = 1 (δv R ) (2.56) x i δv R t If the ensity ρ = constant is inclue, iv(ρu) implies a mass ensity source at the point x i at time t. Corresponingly, rot (U) orcurl(u) representthe vortex ensity of the velocity fiel at the point x i at time t. Ifcurl(U) =0, a flui element at the point x i,anattimet, experiences no contribution

15 2.8 Line, Surface an Volume Integrals 29 to its rotation by the velocity fiel. For rot (U) 0attimet an at point x i, a flui element consequently experiences, at the corresponing point, a contribution to its rotational motion. In summary, the operators escribe above can be formulate as follows: ( ) a gra(a) = = a (2.57) x i iv (U) = U i = U i (2.58) x i a = 2 a = a = ( ) a x i x i = 2 a x i x i (2.59) U j rot (U) = U = ɛ ijk (2.60) x i These operators will be employe for the erivation of the basic equations of flui mechanics an also when ealing with flow problems. 2.8 Line, Surface an Volume Integrals The line integral of a scalar function a(x i,t) along a line S is efine as follows: I s (t) = a S = a(x i,t)s with x i S (2.61) S S Line integrals of this kin are require in flui mechanics to efine the position of the center of gravity of a line. Their computation is carrie out in three stepsasfollowsfort = constant: 1. The viewe curve is parameterize: S : s(γ) ={s i (γ)} T, α γ β (2.62) 2. The arche element s is efine by ifferentiation: (si ) s = s i (γ) 2 γ γ = (γ) γ (2.63) γ 3. The computation of the efine integral from γ = α to γ = β: β (si ) 2 I s = a(s i (γ)) γ I s (2.64) γ α The application of the above steps for the computation of the efine integral leas for a = 1 to the length of the consiere curve s(γ) between γ = α an γ = β.

16 30 2 Mathematical Basics Analogous to the above consierations, the integration of a vector fiel along a curve can be carrie out in the following way: I si (t) = S a i s i = β α a i (s j (γ)) s i(γ) γ at time t (2.65) γ Computations of the work one in the fiels of forces, the circulation of mass an momentum flow in the case of two-imensional flow fiels are effecte via such efine integrations of vector fiels along space lines. Analogous to the integrals along lines or along line segments, space integrals for scalar an vectors can also be efine an compute accoring to the following computation rules: I F (t) = a F at time t (2.66) F If F is the surface area of a consiere flui element an a(x i,t) a scalar fiel, that is continuous on the surface, the above integral as the surface integral is name from a to F. The surface-average value of a is compute as follows: ã = 1 a F 0 (surface mean value) (2.67) F 0 F 0 For the surface integral of a vector fiel hols that I F (t) = a i F i = a i n i F at time t (2.68) F F For the case a i = U i, i.e. the execution of a surface integration over the velocity fiel, an integral value is obtaine that correspons to the instantaneous volume flow through the surface F : Q(t) = U i F i (volume flow through F at time t) (2.69) F Analogously, the mass flow through F is compute by Ṁ(t) = ρu i F i (mass flow through F at time t) (2.70) F The mean mass flow ensity is given by ṁ(t) = 1 F F ρu i F i (2.71)

17 2.9 Integral Laws of Stokes an Gauss 31 The above integrations can be extene to volume integrals, which again can be applie to scalar an vector fiels. If V esignates the volume of a regular fiel an a(x i,t) a steay scalar fiel (occupation function) given in this space, the total occupancy of the space is compute as follows: I V (t) = a V (total occupancy of V through a at time t) (2.72) V The mass of a regular space with the ensity istribution ρ(x i,t)resultsin the following triple integral of the ensity istribution ρ(x i,t): M(t) = ρ(x i,t)x 1 x 2 x 3 (2.73) V Here, variables with an symbols inicate surface- an volume-average quantities in this chapter of the book. For the practical implementation of surface an volume integrations, it is often avantageous to employ the laws of Gulin: 1. Law of Gulin ( ): The surface area of a boy with rotational symmetry is given by s 2π r(s) s, withs enoting an arc element of the plane curve s generating the boy an r(s) enoting the istance of s from the axis of rotation. 2. Law of Gulin ( ): The volume of a boy with rotational symmetry is given by F 2π r s(f ) F,withF enoting an area element of the area enclose by the plane curve s generating the boy an r s (F ) enoting the istance of F from the axis of rotation. 2.9 Integral Laws of Stokes an Gauss The integral law name after Stokes ( ) reas a s = rot a F (2.74) s O s which means that the line integral of a vector a over the entire ege line of a surface is equal to the surface integral of the corresponing rotation of the vector quantity over the surface. Thus the integral law of Stokes represents a generalization of Green s law ( ), which was formulate for plane surfaces, i.e. for spatial areas. If one stretches two ifferent surfaces over a bounary of surface S, Stokes law gives rot a F = rot a F (2.75) O S1 O S2

18 32 2 Mathematical Basics where S is equal to the stretching quantities of O S1 an O S2. If one introuces by Γ = Us the term of circulation of a vector fiel U along a bounary S S employing the mean-value law of the integral calculus from the rot integral of velocity fiel U i over a surface O S with normal n, surface area F an bounary curve S, the surface integral of the vector fiel U when F 0in the borerline case results: 1 Γ = n rot U = lim U S F 0 F This relation makes it clear that the rotation effect of a flui element is at its maximum when the surface normal n is in the irection of the rot U vector. The integral law name after Gauss can be formulate as follows: ai iv a V = V = a F = a i n i F (2.76) x i V V O V Thus the flow of the vector fiel a(x i,t) through the surface of a regular space, i.e. the flow from the insie to the outsie, is equal to the volume integral of the ivergence over the space. The mean-value theorem of the integral calculus for V 0 an consieration of a velocity fiel U i give iv U = U i 1 = lim U F (2.77) x i V 0 V The ivergence of a velocity fiel thus measures the flow emerging from the volume unit, i.e. it is the source ensity of U i in the point x i at time t. O V S O V 2.10 Differential Operators in Curvilinear Orthogonal Coorinates The compilation of important equations an efinitions of vector analysis to ate is base on the Cartesian coorinate system. A great number of problems can, however, be treate more easily in a curvilinear coorinate system usually aapte to a consiere special geometry. As examples are cite the creeping flow aroun a sphere or the flow through a tube with a circular cross-section which can be escribe appropriately in spherical an cylinrical coorinates, respectively. In aition, the solution of a flow problem can often be simplifie consierably by exploiting the symmetry properties of the problem in a curvilinear coorinate system aapte to the geometry. In this section, only some frequently use relationships for the ifferential operators in curvilinear orthogonal coorinate systems will be recalle without strict erivations. More etaile an mathematically precise presentation

19 2.10 Differential Operators in Curvilinear Orthogonal Coorinates 33 can be foun in the corresponing literature, as for example [2.3], [2.4] an [2.5], on whose presentation this section is oriente. General curvilinear coorinates (x 1,x 3,x 3 ) can be compute from Cartesian coorinates (x, y, z) by (local) unequivocally reversible relations: x 1 = x 1(x, y, z) x 2 = x 2(x, y, z) (2.78) x 3 = x 3(x, y, z) Conversely the Cartesian coorinates epen on the curvilinear ones: x = x(x 1,x 2,x 3) y = y(x 1,x 2,x 3) (2.79) z = z(x 1,x 2,x 3). If one hols on to two coorinates, one obtains, with the thir coorinate as a free parameter, a space curve, the so-calle coorinate line, for example: Concerning the respective tangential vectors r = r(x 1,x 2 = a, x 3 = b) (2.80) t i = r x, i =1, 2, 3 (2.81) i of the coorinate lines in point P (x 1,x 2,x 3 ) with the efinition of the socalle metric coefficients h i := r, i =1, 2, 3 (2.82) the unit vectors x i e i = 1 h i t i, i =1, 2, 3 (2.83) can be efine that form the basic vectors for a local reference system in point P. To be emphasize here is the local character of this reference system for general curvilinear coorinates, as the basic vectors themselves can epen on coorinates, contrary to coorinate-inepenent basic vectors (e x, e y, e z ) of the Cartesian coorinate system. If the coorinate lines at each point stan vertically on one another in pairs, i.e. the following relationship e i e j = δ ij (2.84) hols, one esignates the coorinate system curvilinear orthogonal coorinate system. Curvilinear orthogonal coorinate systems are the subject of this section. The reaer intereste in general curvilinear coorinate systems is referre, for example, to the book by R. Aris [2.3].

20 34 2 Mathematical Basics If one consiers two infinitesimal closely neighbouring points P 1 (x 1,x 2,x 3) an P 2 (x 1 +x 1,x 2 +x 2,x 3 +x 3 ), for the ifference of their position vectors r 1 = r(x 1,x 2,x 3 )anr 2 = r(x 1 +x 1,x 2 +x 2,x 3 +x 3 )atthe lowest orer (Taylor expansion) 3 r r = r 2 r 1 = x i (2.85) x i=1 i hols. The length of the istance vector r, the so-calle line element s, when employing the efinition for the metric coefficients, is given by 3 s 2 =r 2 = h 2 i x2 i (2.86) A vector fiel f is represente by its components in curvilinear coorinate systems: f = f 1 e 1 + f 2 e 2 + f 3 e 3 (2.87) Without erivation, the following relationships for ifferential operators are state in curvilinear orthogonal coorinates: Surface elements: ( r S = x i i=1 ) r x x i x j, i j =1, 2, 3 (2.88) j Volume elements: V = h 1 h 2 h 3 x 1x 2x 3 (2.89) Graient of a scalar fiel Φ: 3 1 Φ gra Φ = Φ = h i=1 i x e i (2.90) i Divergence: ( 1 h2 h 3 f 1 iv f = f = h 1 h 2 h 3 x + h 1h 3 f 2 1 x + h ) 1h 2 f 3 2 x (2.91) 3 Rotation: h 1 e 1 h 2 e 2 h 3 e 3 1 rot f = f = h 1 h 2 h 3 x 1 x 2 x (2.92) 3 h 1 f 1 h 2 f 2 h 3 f 3 Laplace operator: Φ = Φ = iv gra Φ = [ ( ) 1 h2 h 3 Φ h 1 h 2 h 3 x 1 h 1 x + ( ) h1 h 3 Φ 1 x 2 h 2 x + ( )] h1 h 2 Φ 2 x 3 h 3 x 3 (2.93)

21 2.10 Differential Operators in Curvilinear Orthogonal Coorinates 35 When employing ifferential operators in curvilinear coorinates, the epenence of the (local) unit vectors an metric coefficients of the coorinates is also to be taken into account at least in principle. Example 1: Cylinrical Coorinates (r, ϕ, z) (Fig.2.6) Conversion in Cartesian coorinates: x = r cos ϕ y = r sin ϕ (2.94) z = z (0 r<, 0 ϕ 2π, <z< ) Position vector: Local unit vectors: r = x(r, ϕ, z)e x + y(r, ϕ, z)e y + z(r, ϕ, z)e z = rr ρ (ϕ)+ze z (2.95) Metric coefficients or scaling factors: Graient: e r =cosϕe x +sinϕe y e ϕ = sin ϕ e x +cosϕe y (2.96) e z = e z h r =1, h ϕ = r, h z = 1 (2.97) gra Φ = Φ r e r + 1 Φ r ϕ e ϕ + Φ z e z (2.98) x 3 Fig. 2.6 Cylinrical coorinates x 1 y r. z x x 2

22 36 2 Mathematical Basics z y x Fig. 2.7 Spherical coorinates Example 2: Spherical Coorinates (r, θ, φ) (Fig.2.7) Conversion in Cartesian coorinates: x = r sin θ cos φ y = r sin θ sin φ (2.99) z = r cos θ (0 r<, 0 θ π, 0 φ 2π) Metric coefficients or scaling factors: h r =1, h θ = r, h φ = r sin θ (2.100) 2.11 Complex Numbers The introuction of complex numbers permits the generalization of basic mathematical operations, as for example the square rooting of numbers, so that the extene grouping of numbers can be state as follows: Complex numbers Real numbers Rational numbers Integer numbers Positive integer numbers (Natural numbers) Zero Imaginary numbers Irrational numbers Fractional numbers Negative integer numbers By extening to complex functions, mathematically interesting escriptions of technical problems become possible, for example the entire fiel of potential

23 2.11 Complex Numbers 37 flows; see Chap. 10. Complex numbers an complex functions therefore have an important role in the fiel of flui mechanics. As will be shown, potential flows can be ealt with very easily through functions of complex numbers. It is therefore important to provie here an introuction to the theory of complex numbers in a summarize way Axiomatic Introuction to Complex Numbers A complex number can formally be introuce as an arrange pair of real numbers (a, b) where the equality of two complex numbers z 1 =(a, b) an z 2 =(c, ) isefineasfollows: Equality: z 1 =(a, b) =(c, ) =z 2 hols exactly when a = c an b = hols, where a, b, c, R. The first component of a pair (a, b) is name the real part an the secon component the imaginary part. For b =0,z =(a, 0) is obtaine, with the real number a, sothatall the real numbers are a sub-set of the complex numbers. When etermining basic arithmetics operations, one has to keep in min that operations with complex numbers lea to the same results as in the case of arithmetics of real numbers, provie that the operations are restricte to real numbers in the above sense, i.e. z =(a, 0). Aitions an multiplications of complex numbers are introuce by the following relationships: Aition: (a, b)+(c, ) =(a + c, b + ) Multiplication: (a, b) (c, ) =(ac b, a + bc) (2.101) Then, (a, 0) + (c, 0) = (a + c, 0) = a + c (a, 0) (c, 0) = (ac, 0) = ac (2.102) i.e. no contraictions to the computational rules with real numbers arise. The quantity of the complex numbers (enote C in the following) is complete as far as aition an multiplication are concerne, i.e. with z 1,z 2 C follows: z 3 = z 1 + z 2 C z 3 = z 1 z 2 C (2.103) Furthermore, it can be shown that the above operations of aition an multiplication satisfy the following laws: Commutative concerning aition: z 1 + z 2 = z 2 + z 1 Commutative concerning multiplication: z 1 z 2 = z 2 z 1

24 38 2 Mathematical Basics Associative concerning aiton: (z 1 + z 2 )+z 3 = z 1 +(z 2 + z 3 ) Associative concerning multiplicaton: (z 1 z 2 )z 3 = z 1 (z 2 z 3 ) Distributive properties: (z 1 + z 2 )z 3 = z 1 z 3 + z 2 z 3 Analogous to the case of the real number z(a, 0), a so-calle purely imaginary number can also be introuce: z =0,b. A complex number z =(a, b) is calle imaginary if a =0anb 0. Moreover, one puts i =(0, 1) an calls i an imaginary unit. Accoring to the multiplication rules, introuce for complex numbers, this complex number i, i.e. the number pair (0, 1), has a special role, namely, i 2 =(0, 1) (0, 1) = ( 1, 0) = 1 (2.104) i.e. the multiplication of the imaginary unit number by itself yiels the real number 1. Base on equation (2.103), it can be represente as i = 1 (2.105) where the unambiguity of the root relationship for i requires some special consierations. Because z =(a, b) =(a, 0) + (0,b)=(a, 0) + (0, 1) (b, 0) = a + ib (2.106) each complex number z =(a, b) can also be written as the sum of a real number a an an imaginary number ib. Subtraction an ivision can be achieve by inversion of the aition an multiplication, i.e., (z 1 z 2 ) is equal to the complex number z 3,forwhich z 2 + z 3 = z 1 (2.107) hols. Following the above notation with z 1 =(a, b), z 2 =(c, ) resultsin z 1 z 2 =(a c, b ) (2.108) ( ) z 1 ac + b = z 2 (c ), bc a (c (2.109) ) In the above presentations, elementary mathematical operations base on the quantity C of the complex numbers were introuce. All other properties of the complex numbers are followe in the implementation of these efinitions Graphical Representation of Complex Numbers In orer to explain the above properties of complex numbers, they are often shown graphically in ways summarize below. Several kins of presentations are chosen in the literature for a better unerstaning.

25 2.11 Complex Numbers 39 iy iy Every point z in the plane represents a complex number z=x+iy Fig. 2.8 Diagram of a complex number in the Gauss number plane x x The Gauss Complex Number Plane As the complex number z = x + iy represents an arrange pair of numbers, a rectangular coorinate system is recommene for the graphical representation of complex numbers, in which a real axis for x an an imaginary axis for iy is efine. The complex number z = x + iy is then efine as a point in this plane, or as a vector z from the origin of the coorinate system to the point Z with the coorinates (x, iy). This is illustrate in Fig. 2.9, where the aition an subtraction of complex numbers are state graphically Trigonometric Representation If one consiers the graphical representation in Fig. 2.8, the following trigonometric relations can be given for complex numbers: x = r cos ϕ an y = r sin ϕ with r = z (2.110) A complex number can therefore be written as follows: or z = r cos ϕ + i(r sin ϕ) =r(cos ϕ + i sin ϕ) (2.111) z = re iϕ (2.112) The connection between the exponential function an the trigonometric functions follows immeiately by a series expansion of the exponential function an rearrangement of the series, i.e. e iϕ (iϕ) k = k! k=0 = k=0 ( 1) k ϕ2k (2k)! + i ( 1) k ϕ2k+1 =cosϕ + i sin ϕ (2k +1)! k=0 (2.113)

26 40 2 Mathematical Basics Fig. 2.9 Diagram of the aition an subtraction of complex numbers in the Gauss number plane Fig Graphical representation of multiplication of complex numbers With the following relationship, the multiplication an ivision of complex numbers can be carrie out: z 1 z 2 = r 1 r 2 e i(ϕ1+ϕ2) = r 1 r 2 (cos(ϕ 1 + ϕ 2 )+isin(ϕ 1 + ϕ 2 )) (2.114) z 1 = r 1 e i(ϕ1 ϕ2) = r 1 (cos(ϕ 1 ϕ 2 )+isin(ϕ 1 ϕ 2 )) (2.115) z 2 r 2 r 2 These multiplications an ivisions of complex numbers can be represente graphically as shown in Figs. 2.9 an At this point, it is avisable to iscuss the treatment of the roots of complex numbers. It is explaine below, how the mathematical operator n () is to be applie to a complex number. It is agree that n z (n N = (natural number)) is the set of all those numbers raise to the 1/nth power of the number z. Therefore, if one puts z = r(cos ϕ + i sin ϕ) (2.116) Then n z = n r ( cos ϕ+2kπ n ) + i sin ϕ+2kπ n = n re i( ϕ n + 2kπ n ) k =0, 1, 2,...,n 1 (2.117)

27 2.11 Complex Numbers 41 Fig Graphical representation of ivision of complex numbers n i.e. z is a set of complex numbers consisting of n numbersofvaluesthat can be interprete geometrically in the complex plane as corner points of a polynomial, which is inscribe in a circle with raius n r aroun the zero point. Specifically for k = 0, for example: n ( z = n r cos ϕ n + i sin ϕ ) = n re i ϕ n (2.118) n Stereographic Projection The above representations of complex numbers were escribe by using the plane employe in the fiel of analytical geometry an well known trigonometric relationships were use. For many purposes it proves more favorable to unerstan the points in the x iy plane as projections of points lying on a unit sphere, whose poles lie on the axis perpenicular to the complex plane. One of the poles of the sphere lies at the zero point, whereas the other takes the position coorinates (0, 0, 1). Stereographic projections are carrie out from the latter pole as inicate in Fig Thus each point of the plane correspons precisely to a point of the sphere which is ifferent from N an vice versa, i.e. the spherical surface is, apart from the starting point of the projection, projecte reversibly in an unequivocal manner on to the complex plane. The figure is circle-allie an angle-preserving. The property of the circle-allie figure inicates that each circle on the sphere is projecte as a circle or a straight line on the plane (an vice versa). The angle-preserving figure signifies that two arbitrary circles (an generally any two curves on the sphere) intersect at the same angle as their stereographic projection in the plane (an vice versa).

28 42 2 Mathematical Basics Fig Representation of the stereographic projection (complex sphere of Riemann) Elementary Function Complex functions are efine analogously to the introuction of real functions an can be given as follows: When C is an arbitrary set of complex numbers, C can be esignate as the omain of the complex variables z. If one assigns to each complex variable z, within the omain C, a complex quantity F (z), then F (z) isesignateasthe function of complex variables. The function F (z) represents again a complex quantity: F (z) =Φ + iψ (2.119) Here it is to be consiere in general that the quantities Φ an Ψ again epen on x an iy, i.e. on the coorinates of the complex variable z. When the efinition of a complex function is compare with the often easier unerstanable real functions, the ifferentiation of a complex function has a significant ifference compare to real functions. The existence of a erivative f (x) of a real function f(x) oes not say anything about the existence of possible higher-orer erivatives, whereas from the existence of first-orer erivative f (x) of a complex function F (z), if automatically follows the existence of all higher erivatives, i.e. When a function F (z), in a fiel G C is holomorphic (i.e. istinguishable in a complex manner) an exists an if the function posseses F (z), then there exist all higher-orer erivatives F (z), F (z),... also. Instea of the term holomorphic, the term analytical is often use. The representation of F (z) is often also treate as conformal mapping. The reason for this is base on the fact that, uner certain restrictive conitions, the function F (z) assigne to each point P in the plane z can map into another complex plane as a point Q in an imaginary plane W.Inorer to achieve this unequivocal assignment, a branch of an equivocal function is

29 2.11 Complex Numbers 43 often introuce as the main branch an only the latter is use for computing. The most important complex functions are as follows, see also refs. [2.2] an [2.7]. Polynomials of nth Orer F (z) =a 0 + a 1 z + a 2 z a n z n (2.120) where a 0, a 1,..., a n are complex constants an n a positive total number. The transformation F (z) =az + b is esignate as a linear transformation in general. Rational Algebraic Function F (z) = P (z) (2.121) Q(z) where P (z) anq(z) are polynomials of arbitrary orer. The special case F (z) = az + b (2.122) cz + where a bc 0 is often esignate as a fractional linear function. Exponential Function F (z) =e z =exp(z) (2.123) where e = represents the basis of the (real) natural logarithm. Complex exponential functions have properties that are similar to those for real exponential functions. For example: Trigonometric Functions e z1 e z2 = e (z1+z2) (2.124) e z1 /e z2 = e (z1 z2) (2.125) The trigonometric functions for complex numbers are efine as follows: sin z = eiz e iz 2i cos z = eiz + e iz 2 (2.126) sec z = 1 cos z = 2 e iz + e iz csc z = 1 sin z = 2i e iz e iz (2.127) tan z = sin z cos z = eiz e iz i(e iz + e iz ) cot z = cos z sin z = i(eiz + e iz ) e iz e iz (2.128)

30 44 2 Mathematical Basics Many of the properties of the above functions are similar to those of real trigonometric functions. Thus it can be shown that sin 2 z +cos 2 z =1;1+tan 2 z =sec 2 z;1+cot 2 z =csc 2 z (2.129) sin( z) = sin z; cos( z) = cos z; tan( z) = tan z (2.130) sin(z 1 ± z 2 )=sinz 1 cos z 2 ± cos z 1 sin z 2 (2.131) cos(z 1 ± z 2 )=cosz 1 cos z 2 sin z 1 sin z 2 (2.132) tan(z 1 ± z 2 )= tan z 1 ± tan z 2 1 tan z 1 tan z 2 (2.133) Hyperbolic Functions The hyperbolic functions in the complex case are efine as follows: sinh z = ez e z 2 cosh z = ez + e z 2 (2.134) sech z = 1 cosh z = 2 e z + e z csch z = 1 sinh z = 2 e z e z (2.135) tanh z = sinh z cosh z = ez e z e z + e z For these functions, the following relations apply: coth z = cosh z sinh z = ez + e z e z e z (2.136) cosh 2 z sinh 2 z =1;1 tanh 2 z =sech 2 z;coth 2 z 1 = csch 2 z (2.137) sinh( z) = sinh z; cosh( z) = cosh z; tanh( z) = tanh z (2.138) sinh(z 1 ± z 2 )=sinhz 1 cosh z 2 ± cosh z 1 sinh z 2 (2.139) cosh (z 1 ± z 2 )=coshz 1 cosh z 2 ± sinh z 1 sinh z 2 (2.140) tanh(z 1 ± z 2 )= tanh z 1 ± tanh z 2 (2.141) 1 ± tanh z 1 tanh z 2 From the above relations for trigonometric functions an hyperbolic functions, the following connections can be inicate: sin iz = i sinh z cos iz =coshz tan iz = i tanh z (2.142) sinh iz = i sin z cosh iz =cosz tanh iz = i tan z (2.143)

31 2.11 Complex Numbers 45 Logarithmic Functions As in the real case, the natural logarithm is the inverse function of the exponential function, i.e. it hols for complex cases that F (z) =lnz =lnr + i(ϕ +2kπ) k =0, ±1, ±2,... (2.144) where z = re iϕ hols. It appears that the natural logarithm represents a non-equivocal function. By limitation to the so-calle principal value of the function, an equivocalness can be prouce. Here a certain arbitrariness is given. It can be eliminate by a specially esire branch, on which the equivocalness is guarantee, which is also inicate, for example by (ln z) 0. The logarithmic functions can be efine for any real basis, i.e. also for values that iffer from e. This means that the following can be state: F (z) =log a z z = a F (2.145) where a>0 as well as a 0 an a 1 (2.146) Inverse Trigonometric Functions Inverse trigonometric functions for complex numbers can be state as follows. These functions also are efine as non-equivocal, but show a perioicity: sin 1 z = i ln (iz + ( ) 1 z 2 csc 1 z = 1 i ln i + ) z 2 1 (2.147) z ( cos 1 z = i ln z + ) z 2 1 tan 1 z = 1 ( ) 1+iz 2i ln 1 iz ( sec 1 z = 1 i ln 1+ ) 1 z 2 (2.148) z cot 1 z = 1 ( ) iz +1 2i ln iz 1 (2.149) Inverse Hyperbolic Functions Analogous to the consierations of the trigonometric functions, the inverse functions of the hyperbolic functions can be formulate. These are as follows: ( sinh 1 z =ln z + ( ) z 2 +1 csch 1 i + ) z z =ln 2 +1 (2.150) z ( cosh 1 z =ln z + ) z 2 1 ( sech 1 1+ ) 1 z z =ln 2 (2.151) z

32 46 2 Mathematical Basics tanh 1 z = 1 ( ) 1+z 2 ln coth 1 z = 1 ( ) z +1 1 z 2 ln (2.152) z 1 Differentiation of Complex Functions (Cauchy Riemann Equations) If the function F (z) inafielg C is efine an the limiting value F F (z + z) F (z) (z) = lim (2.153) z 0 z is inepenent of the approximation z 0, then the function F (z) inthe fiel G is esignate analytically. A necessary conition so that the function F (z) =Φ + iψ represents a function analytically in G C is set by the Cauchy Riemann ifferential equations: Φ x = Ψ y Φ y = Ψ x (2.154) When the partial erivations of the Cauchy Riemann equations in G are steay, then the Cauchy Riemann equations are a sufficient conition to say that F (z) is analytical in the fiel G. From the Cauchy Riemenn relations, it can be erive by ifferentiation that the real an imaginary parts of the function F (z), i.e. the quantities Φ(x, y) an Ψ(x, y), fulfill the Laplace equation, i.e. 2 Φ x 2 2 Ψ x Φ y 2 = 0 (2.155) 2 Ψ y 2 = 0 (2.156) Differentiation of Complex Functions If F (z), G(z) anh(z) are analytical functions of the complex variable z, then the ifferentiation laws of the functions result as inicate below. It is easy to see that they are analogus to the function of real variables. z [F (z)+g(z)] = z F (z)+ z G(z) =F (z)+g (z) (2.157) z [F (z) G(z)] = z F (z) z G(z) =F (z) G (z) (2.158) z [cf (z)] = c z F (z) =cf (z), with c as arbitrary constant (2.159) z [F (z)g(z)] = F (z) z G(z)+G(z) z F (z) =F (z)g (z)+g(z)f (z) (2.160)